An Invitation to Noncommutative Geometry
This book is a collection of written versions of lecture series presented at the ‘Workshop on non-commutative geometry’ held in Teheran in 2005. The two longest ones are general review papers on non-commutative geometry. The first one, written by A. Connes and M. Marcolli, starts with a description of a general strategy of how to deal with problems in non-commutative geometry, followed by a long list of important examples of the theory. The second one (by M. Khalkhali) treats operator algebras, topological K-theory and cyclic cohomology. A lecture by B. Noohi on derived and triangulated categories shows why the language of categories is important for non-commutative geometry. Renormalization of non-commutative QFT is discussed by H. Grosse and R. Wulkenhaar, complex tori and spherical manifolds are described in lectures by J. Plazas and non-commutative algebraic geometry is treated by S. Mahanta. The paper by R. J. Szabo explains a role of noncommutative field theories in the theory of D-branes in a type II superstring context. There is also a paper by G. Landi and W. D. Suilekom on non-commutative bundles and instantons. The book offers a very valuable collection of material on an important and quickly developing field of mathematics.